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arXiv:cond-mat/9212013 · Physical review. B, Condensed matter

Bosonic mean-field theory of the spiral phases of Heisenberg antiferromagnets on a chain

Sumathi Rao, Diptiman Sen

Published Nov 1, 1993• 9 citations
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Abstract

We develop a bosonic mean-field theory to describe the spiral phases of a Heisenberg antiferromagnet on a one-dimensional chain, in terms of three bosons at each site. The ground state is disordered, and for large values of the spin S, two different and exponentially small energy gaps are found. The spin-spin correlation function is computed and is shown to decay exponentially at large distances. Our mean-field theory is also shown to be exact in a large-N generalization.

DOI
10.1103/physrevb.48.12763

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We develop a bosonic mean-field theory to describe the spiral phases of a Heisenberg antiferromagnet on a one-dimensional chain, in terms of three bosons at each site. The ground state is disordered, and for large values of the spin S, two different and exponentially small energy gaps are found. The spin-spin correlation function is computed and is shown to decay exponentially at large distances. Our mean-field theory is also shown to be exact in a large-N generalization.
Using the exact diagonalization technique, we study the properties of the ground state of a spin- antiferromagnetic Heisenberg model for a zigzag polymer chain with side radicals connected to the even sites. We consider the nearest-neighbour exchange J and the next-nearest-neighbour exchange ?J along the main chain, and J1 between the even site on the main chain and the radical site. For small ? the ground state is ferrimagnetic. For ?>?c 1, the ground state is a spiral phase, which is characterized by a peak of the static structure factor S(q) locating at an incommensurate value qmax. For ?>?c 2, the ground state is antiferromagnetic. With increasing J1, ?c 1 decreases while ?c 2 has a maximum at about J1 = 0.5. For very small J1 and ? = 0.5, the spin configuration on the main chain is a product of nearest-neighbour singlets. In the antiferromagnetic phase, if J1 is large enough the even site and the radical site form a singlet with exchange-decoupling from the odd site while the odd sites approximately form an antiferromagnetic chain.
The Schwinger-boson mean-field theory is applied to the quantum ferrimagnetic Heisenberg chain. There is a ferrimagnetic long-range order in the ground state. We observe two branches of the low-lying excitation and calculate the spin reduction, the gap of the antiferromagnetic branch, and the spin fluctuation at T=0K.T=0 \mathrm{K}.T=0K. These results agree with the established numerical results quite well. At finite temperatures, the long-range order is destroyed because of the disappearance of the Bose condensation. The thermodynamic observables, such as the free energy, magnetic susceptibility, specific heat, and the spin correlation at T>0K,T>0 \mathrm{K},T>0K, are calculated. The $T{\ensuremath{\chi}}_{\mathrm{uni}}$ has a minimum at intermediate temperatures and the spin-correlation length behaves as ${T}^{\ensuremath{-}1}$ at low temperatures. These qualitatively agree with the numerical results and the difference is small at low temperatures.
Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin-correlation function 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.
We study the antiferromagnetic spin chain with both dimerization and frustration. The classical ground state has three phases, a Néel phase, a spiral phase and a collinear phase, which we study through a nonlinear sigma-model approach. In the spiral phase, the field theory becomes and Lorentz invariant at long distances, a model which is exactly solvable. The low-energy spectrum is doubly degenerate with massive `elementary' spin-1/2 particles and `two-particle' triplet and singlet physical excitations. The field theory also supports solitons which lead to a double degeneracy of the ground state for half-integer spins (when there is no dimerization).
The Heisenberg antiferromagnetic spin chain with both dimerization and frustration is studied. The classical ground state has three phases (a Neel phase, a spiral phase and a colinear phase), around which a planar spin-wave analysis is performed. In each phase, we discuss a non-linear sigma model field theory describing the low energy excitations. A renormalization group analysis of the SO(3) matrix-valued field theory of the spiral phase leads to the conclusion that the theory becomes SO(3)×SO(3)SO(3) \times SO(3)SO(3)×SO(3) and Lorentz invariant at long distances. This theory is analytically known to have a massive spin-1/2 excitation. We also show that Z2 Z_2 ~Z2​  solitons in the field theory lead to a double degeneracy in the spectrum for half-integer spins.
We compute the ground-state spin-spin correlation function for the one-dimensional antiferromagnetic Heisenberg model of spin 1/2, 1, and 3/2. The spin-1 chain has a disordered ground state with a correlation length of 6.2 lattice spacings whereas the spin-1/2 and the spin-3/2 chains both have (ln(r)${)}^{\mathrm{\ensuremath{\sigma}}}$/r decaying correlation functions. The logarithmic corrections are different for the spin-1/2 and the spin-3/2 chains. Finite-size scaling is used to analyze the data for chains of up to 128 spins.